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Mirrors > Home > MPE Home > Th. List > fzsdom2 | Structured version Visualization version GIF version |
Description: Condition for finite ranges to have a strict dominance relation. (Contributed by Stefan O'Rear, 12-Sep-2014.) (Revised by Mario Carneiro, 15-Apr-2015.) |
Ref | Expression |
---|---|
fzsdom2 | ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → (𝐴...𝐵) ≺ (𝐴...𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelz 11889 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 ∈ ℤ) | |
2 | 1 | ad2antrr 764 | . . . . . 6 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐵 ∈ ℤ) |
3 | 2 | zred 11674 | . . . . 5 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐵 ∈ ℝ) |
4 | eluzel2 11884 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐴 ∈ ℤ) | |
5 | 4 | ad2antrr 764 | . . . . . 6 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐴 ∈ ℤ) |
6 | 5 | zred 11674 | . . . . 5 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐴 ∈ ℝ) |
7 | 3, 6 | resubcld 10650 | . . . 4 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → (𝐵 − 𝐴) ∈ ℝ) |
8 | simplr 809 | . . . . . 6 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐶 ∈ ℤ) | |
9 | 8 | zred 11674 | . . . . 5 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐶 ∈ ℝ) |
10 | 9, 6 | resubcld 10650 | . . . 4 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → (𝐶 − 𝐴) ∈ ℝ) |
11 | 1red 10247 | . . . 4 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 1 ∈ ℝ) | |
12 | simpr 479 | . . . . 5 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐵 < 𝐶) | |
13 | 3, 9, 6, 12 | ltsub1dd 10831 | . . . 4 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → (𝐵 − 𝐴) < (𝐶 − 𝐴)) |
14 | 7, 10, 11, 13 | ltadd1dd 10830 | . . 3 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → ((𝐵 − 𝐴) + 1) < ((𝐶 − 𝐴) + 1)) |
15 | hashfz 13406 | . . . 4 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (♯‘(𝐴...𝐵)) = ((𝐵 − 𝐴) + 1)) | |
16 | 15 | ad2antrr 764 | . . 3 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → (♯‘(𝐴...𝐵)) = ((𝐵 − 𝐴) + 1)) |
17 | 3, 9, 12 | ltled 10377 | . . . . . 6 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐵 ≤ 𝐶) |
18 | eluz2 11885 | . . . . . 6 ⊢ (𝐶 ∈ (ℤ≥‘𝐵) ↔ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐵 ≤ 𝐶)) | |
19 | 2, 8, 17, 18 | syl3anbrc 1429 | . . . . 5 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐶 ∈ (ℤ≥‘𝐵)) |
20 | simpll 807 | . . . . 5 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐵 ∈ (ℤ≥‘𝐴)) | |
21 | uztrn 11896 | . . . . 5 ⊢ ((𝐶 ∈ (ℤ≥‘𝐵) ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → 𝐶 ∈ (ℤ≥‘𝐴)) | |
22 | 19, 20, 21 | syl2anc 696 | . . . 4 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐶 ∈ (ℤ≥‘𝐴)) |
23 | hashfz 13406 | . . . 4 ⊢ (𝐶 ∈ (ℤ≥‘𝐴) → (♯‘(𝐴...𝐶)) = ((𝐶 − 𝐴) + 1)) | |
24 | 22, 23 | syl 17 | . . 3 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → (♯‘(𝐴...𝐶)) = ((𝐶 − 𝐴) + 1)) |
25 | 14, 16, 24 | 3brtr4d 4836 | . 2 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → (♯‘(𝐴...𝐵)) < (♯‘(𝐴...𝐶))) |
26 | fzfi 12965 | . . 3 ⊢ (𝐴...𝐵) ∈ Fin | |
27 | fzfi 12965 | . . 3 ⊢ (𝐴...𝐶) ∈ Fin | |
28 | hashsdom 13362 | . . 3 ⊢ (((𝐴...𝐵) ∈ Fin ∧ (𝐴...𝐶) ∈ Fin) → ((♯‘(𝐴...𝐵)) < (♯‘(𝐴...𝐶)) ↔ (𝐴...𝐵) ≺ (𝐴...𝐶))) | |
29 | 26, 27, 28 | mp2an 710 | . 2 ⊢ ((♯‘(𝐴...𝐵)) < (♯‘(𝐴...𝐶)) ↔ (𝐴...𝐵) ≺ (𝐴...𝐶)) |
30 | 25, 29 | sylib 208 | 1 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → (𝐴...𝐵) ≺ (𝐴...𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 class class class wbr 4804 ‘cfv 6049 (class class class)co 6813 ≺ csdm 8120 Fincfn 8121 1c1 10129 + caddc 10131 < clt 10266 ≤ cle 10267 − cmin 10458 ℤcz 11569 ℤ≥cuz 11879 ...cfz 12519 ♯chash 13311 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-card 8955 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-n0 11485 df-xnn0 11556 df-z 11570 df-uz 11880 df-fz 12520 df-hash 13312 |
This theorem is referenced by: irrapxlem1 37888 |
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